Presentation on theme: "example: four masses on springs"— Presentation transcript:
1 example: four masses on springs Normal Modesexample: four masses on springs
2 Four masses on springsFind a physical description of a system that might look like this:We will use:Newton’s lawsVectorsMatricesLinearity (superpositions)Complex numbersDifferential equationsExponential functionsEigenvaluesEigenvectors
3 Problem: masses on springs (I) We consider four masses connected to springs with spring constant k and their motion restricted to one spatial dimension.Step 1: Write down Newton’s law for the motion of the masses
4 Problem: masses on springs (I) Step 2: Combine the degrees of freedom into a vector and write the equations of motion as a matrix equationStep 3: Use a complex exponential as the ansatz for the solution to this equation
5 Problem: masses on springs (II) Step 4: Substitute this ansatz into the equation of motionStep 5: Solve the eigenvalue equation for eigenvalues 2 and eigenvectors vWe do not need the negative frequency solutions since we only consider the real part as physically relevant
6 Problem: masses on springs (III) Find the eigenvectors (here not normalized) from the corresponding homogeneous equationsStep 6: The general solution is then given by a superposition of all these normal modes with complex amplitudes A1, A2, A3, A4 chosen to meet the initial conditions:If the system is in one of these normal modes (i.e. all Ai zero except An) all masses will oscillate with the same frequency n=n(k/m)1/2 and constant amplitude ratios defined by vn .
7 Problem: masses on springs (IV) Visualization of the normal modes